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Analytical description of the distributions of primary and secondary cosmogenic particles

Shvetaank Tripathi, Prashant Shukla

TL;DR

The paper addresses the need for simple analytic forms that capture the energy and angular distributions of cosmogenic particles. It introduces modified power-law distributions with a low-energy energy-loss term for primaries, an exponential decay term for muons, and a neutrino-specific source term, all tied to zenith-angle dependence via $\cos^{n-1}\theta$. The authors derive explicit formulas for primaries $I(E)=I_0 N\left(\frac{E_1}{E}+E\right)^{-n}$, muons $I(E)=I_0 N\left(\frac{E_{0d}}{\cos\theta}+E\right)^{-n}\exp\left(-\frac{k}{E\cos\theta}\right)$, and neutrinos $I(E)=I_0 N\left[\frac{E_0 d}{\cos\theta}+E\right]^{-n}\left(1+\frac{E}{\epsilon}\right)^{-1}$, with fits to data and simulations yielding $n$ in the 2.7–3.0 range for hadronic primaries and muons, and flavor-dependent values for neutrinos. These analytic forms enable fast integrated-flux calculations and practical detector simulations, while remaining physically interpretable. The framework is extensible to additional cosmogenic species (e.g., charm-induced leptons, high-energy photons, heavier nuclei) and can accommodate geomagnetic cutoffs through fitted parameters, making it valuable for both predictive modeling and data-driven extrapolations.

Abstract

In this work, we present an analytical description of the energy distributions of primary and secondary cosmogenic particles on Earth in terms of parameters having clear physical meaning. A modified power law is assumed for energy distributions, incorporating terms such as energy loss/decay, which are effective at low energies, and a source term, which is dominant at high energies. The parametrizations of the momentum distribution of primary protons and helium have been obtained including energy loss term. For muons, both the energy loss and decay terms have been included. It is shown analytically that zenith angle distributions is given by $\cos^{n-1}θ$ in terms of energy index $n$ and the presence of decay term does not affect it. The analytical function describes the muon momentum distribution data at different altitudes and zenith angles. The same form is also applied to describe the atmospheric muon and electron-type neutrino distributions simulated at various sites. The presented analytical functions provide an excellent description of all kinds of cosmogenic particles.

Analytical description of the distributions of primary and secondary cosmogenic particles

TL;DR

The paper addresses the need for simple analytic forms that capture the energy and angular distributions of cosmogenic particles. It introduces modified power-law distributions with a low-energy energy-loss term for primaries, an exponential decay term for muons, and a neutrino-specific source term, all tied to zenith-angle dependence via . The authors derive explicit formulas for primaries , muons , and neutrinos , with fits to data and simulations yielding in the 2.7–3.0 range for hadronic primaries and muons, and flavor-dependent values for neutrinos. These analytic forms enable fast integrated-flux calculations and practical detector simulations, while remaining physically interpretable. The framework is extensible to additional cosmogenic species (e.g., charm-induced leptons, high-energy photons, heavier nuclei) and can accommodate geomagnetic cutoffs through fitted parameters, making it valuable for both predictive modeling and data-driven extrapolations.

Abstract

In this work, we present an analytical description of the energy distributions of primary and secondary cosmogenic particles on Earth in terms of parameters having clear physical meaning. A modified power law is assumed for energy distributions, incorporating terms such as energy loss/decay, which are effective at low energies, and a source term, which is dominant at high energies. The parametrizations of the momentum distribution of primary protons and helium have been obtained including energy loss term. For muons, both the energy loss and decay terms have been included. It is shown analytically that zenith angle distributions is given by in terms of energy index and the presence of decay term does not affect it. The analytical function describes the muon momentum distribution data at different altitudes and zenith angles. The same form is also applied to describe the atmospheric muon and electron-type neutrino distributions simulated at various sites. The presented analytical functions provide an excellent description of all kinds of cosmogenic particles.
Paper Structure (11 sections, 11 equations, 8 figures, 7 tables)

This paper contains 11 sections, 11 equations, 8 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Flux Distribution of Primary Cosmic Particles
  3. Flux Distribution of Atmospheric Muons
  4. Energy Distribution
  5. Zenith Angle Distribution
  6. Variation in Muon Fluxes with respect to Atmospheric Depth and Zenith Angle
  7. Variation with respect to atmospheric depth :
  8. Variation with respect to zenith angle :
  9. Variation of Integrated flux with respect to atmospheric depth and zenith angle
  10. The Distribution of Atmospheric Neutrinos
  11. Summary and Prospects

Figures (8)

  • Figure 1: Proton and Helium flux MAENO2001121SHIKAZE20071542009BRASP..73..564P2006astro.ph.12377P2011Sci...332...69AAGUILAR2002331PhysRevLett.114.171103PhysRevLett.115.2111012009ApJ...707..593A2010ApJ...714L..89A as a function of momentum, fitted with Eq. \ref{['eq:prohelflux']}.
  • Figure 2: Muon flux as a function of momentum at various locations HAINO200435Rastin:1984nuMGardener_1962PJHayman_1962https://doi.org/10.1029/92JA02672PhysRevD.19.1368PhysRevLett.83.4241Sogarwal:2022kgw, fitted with Eq. \ref{['eq:muonflux']}.
  • Figure 3: The negative muon flux as a function of momentum measured at different atmospheric depths PhysRevD.53.35. The solid lines are obtained by fitting the data with Eq. \ref{['eq:muonflux']} keeping $E_0$ = 0.002 GeV/g-cm$^{-2}$ and $k$ free.
  • Figure 4: The negative muon flux as a function of momentum measured at different atmospheric depths PhysRevD.53.35. The solid lines are obtained by fitting the data with Eq. \ref{['eq:muonflux']} keeping $E_0$ = 0.002 GeV/g-cm$^{-2}$ and $k$ = 0.80 GeV.
  • Figure 5: The total muon flux at different zenith angles ACHARD200415 at an altitude of 450 m is given in right figure. The solid lines are obtained by fitting the data with Eq. \ref{['eq:muonflux']} keeping $E_0$ = 0.005 GeV/g-cm$^{-2}$ and $k$ = 0.80 GeV.
  • ...and 3 more figures